In city traffic yes, it would. Can't speak for you, but I spend most of the time in traffic in primary position.
Its to do with the speed of slower cyclists and just how many more you'd see claiming primary as opposed to cycling close to or on the pavement. Standing in London recently it occurred to me that the only way that the traffic was working at all was by the slower cyclists being right in the gutter, allowing fast and close (basically illegal) overtaking as the norm rather than the exception. If there is traffic on both sides of the road and not sufficient space to safely pass a cyclist in secondary position without running in to oncoming traffic (normal city street) then it is not possible to overtake safely, and the solution is not to move over to the lefta bit more, it is to claim primary. Get all the slower cyclists out in the middle of the lane and a lot of cities suddenly start moving at 11mph, not as a mean speed but as a peak speed.
But it gets tougher than that... Theres a road I use maybe three days in five, its a 30mph road but usually speeded on as it is long and straight. Parked cars always down one side, no chance whatsoever of safe overtaking if you're in primary and there is traffic on the other side, so if you're on the same side of the parked cars you really, really should stay in primary else sooner or later you'll be doored. On average, over the 1100 yards or so of that road, I'll overtake 6 to 8 other cyclists, I'll average probably 18mph along there if I'm taking it easyish, although I've done 32mph down there. Cars will uniformly do 35mph, if every cyclist was in primary at their standard pace (10 -12 mph), avoiding the car doors or simply preventing people from overtaking them dangerously with oncoming traffic... Well, the traffic would be far slower, it would back up to the junction at the end (already does sometimes), few cars would get on to the road when the lights changed (theres always a bike or three...), thus the traffic is now static down on to Milton Road, possibly right back to the roundabout.
Its not that every single road would be blocked, its more that there would be multiple significant new bottlenecks in every town and city, which would require very different management. And, frankly, to a typical motorist its easier to just call that gridlock.